Question
(a) If \( 4^{x}=43 \), then \( x=\square \) (b) If \( 15^{-x}=4 \), then \( x=\square \) Question Help: Video
Ask by Mathis Salinas. in the United States
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
**(a) \( x = \frac{\ln(43)}{\ln(4)} \)
(b) \( x = -\frac{\ln(4)}{\ln(15)} \)
Solution
**(a) Solving \( 4^x = 43 \):**
1. Take the natural logarithm of both sides:
\[
\ln(4^x) = \ln(43)
\]
2. Use the logarithm power rule \(\ln(a^b) = b \ln(a)\):
\[
x \ln(4) = \ln(43)
\]
3. Solve for \( x \):
\[
x = \frac{\ln(43)}{\ln(4)}
\]
**(b) Solving \( 15^{-x} = 4 \):**
1. Take the natural logarithm of both sides:
\[
\ln(15^{-x}) = \ln(4)
\]
2. Apply the power rule:
\[
-x \ln(15) = \ln(4)
\]
3. Solve for \( x \):
\[
x = -\frac{\ln(4)}{\ln(15)}
\]
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Simplify this solution Bonus Knowledge
To solve \( 4^{x} = 43 \), you can take the logarithm of both sides. This leads to \( x = \frac{\log(43)}{\log(4)} \). Calculate using a scientific calculator, and you'll find that \( x \approx 2.426 \). It’s like you're casting a spell to uncover the hidden power of numbers! For \( 15^{-x} = 4 \), you can rewrite it as \( \frac{1}{15^x} = 4 \), which implies that \( 15^x = \frac{1}{4} \). Again, taking logarithms gives \( x = -\frac{\log(4)}{\log(15)} \). Upon calculating, you’ll discover \( x \approx -0.569 \). Solving these is like figuring out a treasure map—you’re just a few calculations away from uncovering the loot!
